Statistics Probability Distribution

Question

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 46 ounces and a standard deviation of 10 ounces.

Use the Empirical Rule, find

a) 99.7% of the widget weights lie between

b) What percentage of the widget weights lie between 26 and 76 ounces?

c) What percentage of the widget weights lie above 36 ? 

Answer 1

The "Empirical Rule" says that 68% of the data falls within plus and minus 1 standard deviation from the mean, 95% falls within plus and minus 2 standard deviations from the mean and 99.7% falls within plus and minus three standard deviations from the mean.

In this case, plus and minus 1 standard deviation would be 46 - 10 (or 36) and 46 + 10 (or 56) meaning 68% of all widgets are between 36 ounces and 56 ounces.

Also, in this case plus and minus 2 standard deviation would be 46 - 20 (or 26) and 46 + 20 (or 66) meaning 95% of all widgets are between 26 ounces and 66 ounces.

Again, in this case plus and minus 3 standard deviation would be 46 - 30 (or 16) and 46 + 30 (or 76) meaning 99.7% of all widgets are between 16 ounces and 76 ounces which is the answer to question a).

To determine the answer to question b), you need to think of half of the standard deviation. Since plus and minus 3 standard deviations is 99.7%, then just plus 3 standard deviations (from 46 to 76) is (99.7%)/2 = 49.85%. And since plus and minus 2 standard deviation is 95% then minus 2 standard deviations (from 26 to 46) is (95%)/2 = 47.5%. So minus 2 to plus 3 standard deviations (from 26 to 76) is 49.85% + 47.5% = 97.35%.

To determine the amount above 36, first consider everything above the mean (46) as 50% of the widgets. Then use half of the 68% (minus one standard deviation) or 34% to get the total from 36 to 46. So the amount above 36 is 34% + 50% = 84%